Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 2x + 43$, and $ m \angle BOC = 4x - 37$, find $m\angle AOB$. $O$ $A$ $C$ $B$
Solution: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 43} + {4x - 37} = {90}$ Combine like terms: $ 6x + 6 = 90$ Subtract $6$ from both sides: $ 6x = 84$ Divide both sides by $6$ to find $x$ $ x = 14$ Substitute $14$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 2({14}) + 43$ Simplify: $ {m\angle AOB = 28 + 43}$ So ${m\angle AOB = 71}$.